Muon catalyzed fusion in the target HT/D2/HD by taking fast muon ...

International Journal of Physical Sciences Vol. 7(15), pp. 2282 - 2286, 9 April, 2012 Available online at http://www.academicjournals.org/IJPS DOI: 10.5897/IJPS11.831 ISSN 1992-1950 ©2012 Academic Journals

Full Length Research Paper

Muon catalyzed fusion in the target H-T/D2/H-D by taking fast muon conversion A. Avazpour1*, R. Gheisari2, E. Rahnama3 and N. M. Sharifloo4 1

Department of Physics, Yasouj University, Yasouj, Iran. Department of Physics, Persian Gulf University, Bushehr, Iran. 3 Department of Physics, Islamic Azad University Teharn North Branch, Tehran, Iran 4 Department of Physics, Imam Hossein University, Tehran, Iran. 2

Accepted 2 April, 2012

In past years, two/three-layer hydrogen targets with a composition of H/D/T at temperature ~3K have been studied for muon catalyzed fusion. Several theoretical and experimental studies have evaluated that the conversion efficiency of fast muons in such targets is very low. Here, the muon cycling coefficient has been determined for the heterogeneous target of H  T / D2 / H  D by taking the conversion efficiency of incident muons of momentum 26.25 MeV / c . This parameter extremely affects the muon cycling coefficient by the reason of slow muon cycles in the third layer of H  D . The optimal muon cycling coefficient equals 87.5. Key words: Three-layer target, condensed hydrogen-deuterium layer, muon cycling coefficient, fast muon.

INTRODUCTION The negative muon is a lepton of the second generation with the mass about 207 times larger than that of electron, and has a mean life time

   0 1  2.198s .

The 0 is the muon decay rate.

The negative muon can tightly be bounded as a muonic atom. Such a muon can participate in a variety of atomic and molecular processes. Muon dynamics in muon catalyzed fusion (CF) includes very complicated processes, such as muonic interactions which can take place in hydrogen, can be recognized as muonic atom formation, elastic scattering, inelastic scattering, muonic molecular formation, fusion and sticking (Mulhauser et al., 2001). A muonic atom during collision with usual atoms and molecules can form a muonic molecule. The latter in turn can result in fusion reactions between nuclei (Froelich, 1992; Markushin, 1994). When the muonic atom x (where x denotes one of the hydrogen

*Corresponding author. E-mail: [email protected]. Tel: +987412223187.

isotopes p, d, t) collides with molecules like H2, HD, DT, T2 or D2, it forms muonic molecules, such as ddμ, ttμ, dtμ, pp, pd and pt. The muon is probably released after nuclear fusion, and can repeat the same processes. This paper considers a multilayered solid target and describes the atomic and nuclear processes. The fast muon conversion and hydrogen solid effect are taken into account for estimation of the muon cycling coefficient in the heterogeneous target H  T / D2 / H  D .

MUON CATALYZED FUSION IN SUGGESTED MULTILAYERED CONDENSED SYSTEM A scheme of the suggested heterogeneous solid target H  T / D2 / H  D is as shown in Figure 1. Muons of

momentum 26.25 MeV / c are injected into the H  T layer. In this layer, after muon stops at a distance of about 0.4 mm, muon cascade reactions will begin (Wozniak et al., 2003; Marshall et al., 1999). Once a negative muon enters the first layer with small admixture of tritium, μp atoms are formed rapidly due to many

Avazpour et al.

t1s

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population. By the help of Ramsauer-Townsend

(RT) effect (Mulhauser et al., 2006),

t1s

atoms escape

from the H-T layer. A thickness of  800m has been chosen for this layer. This muonic atom, during collisions in the D2 layer with  5m thick, behaves as a heavy neutron because it is neutral, very tightly bound and very small as compared to usual molecules. Hence, it makes slowing down in the D2 rapidly. During a collision in the HD layer with a relative density of   1.45 and  1m Figure 1. Muons of momentum 26.25 MeV / c are injected into the multilayered solid target H  T / D2 / H  D . The μt emission is based on the RT effect: (a) the H / T layer adopted with a small amount tritium impurity for production of fast muonic atoms, (b) deuterium moderator, (c) a fusion layer of H  D . The most of released muons by fusion stay in the fusion layer.

thickness, depending on deuterium concentration, a dt muonic molecule can be formed resonantly (Vesman, 1967; Gheisari, 2010):

t1s  D2  [(dt )dee]

(4)

The deuterium/proton concentrations in the H/D layer are shown with the notations cd and cp, respectively. The proton nuclei with a concentration of

c p0  0.999 . The

de-excitation of p to the ground state competes with isotopic muon transfer:

pnl  t , p  pn'l '  t , p

(1)

p1s  t  t1s  p  183eV

(2)

fusion will occur in dt with a rate of

 eff    n    17.6MeV (1  s ) (5) dt    eff    n  17.6MeV (s )

df t

The muon can be released with a probability of 1   s . eff

The n, l (n' , l ' ) are principal and angular momentum quantum numbers, respectively. As the reaction (Equation 2) takes place in the H-T layer with a relative density of 0  1.2 , the μp atoms are de-excited rapidly (  0 is measured relative to the

density of

liquid

hydrogen). In the reaction (Equation 2), the kinetic energy of t1s atom is about 45 eV. After t formation in the H-T layer, muonized protium-tritium (pt) and muonized tritium-tritium (tt) molecules are probably formed:

 pt   pt t  p   tt   t  t  tt 

(3)

By interactions with hydrogen/deuterium will make the following reactions (Markushin, 1995; Filipowicz et al., 2008):

a    H  p  e    a d  e    D   pd p  d  d  p  F  p d  d  p    pd J   F dFd  d  d    dd 

 a is

the muonic atom formation rate, and

being the formation rate of Their corresponding formation rates (  pt , tt ) are 6

about 10 (

f pt

s

-1

(Markushin, 1994). These fusion rates

,  ) are relatively low. Due to very small f tt

concentration of tritium ( ct

dtf   1.1 1012 s 1 :

 0.12% ) and eventually low

molecular formation rates which were mentioned earlier, these processes do not make a significant effect on the

(6)

F  pd ( dd )

pd ( dd ). F denotes the

d atom. The rate 3/ 2 1/ 2 F=3/2(1/2) is  dd (  dd

spin state of the

of dd molecule

formation for

). The muon can

transfer from proton to deuterium by the rate of pd. The fusion of

f pd molecule with a rate of  pd  in the H-D

layer is through the following channels (Filipowicz et al., 2008):

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Int. J. Phys. Sci.

pd J ,

3    2 He    5.5MeV (J ) 3    2 He    5.5MeV ( J )

(7)

where J is the total angular momentum and υ the vibration-quantum number of the pd molecule. The fusion of

f dd molecule with a rate of dd  is through the

following reactions:

 23 He  n    3.3MeV dd     (8)  p  t    4.04MeV f (10.58dd ) dd 

The muon is captured with the effective probability of 0.58 dd , where  dd equals 12%. As the aforementioned, the resonant formation of dt occurs in the last layer. The released muons in the H/D layer can be captured by proton/deuterium nuclei and then form muonic molecules again and stay in the same layer with very high probability (Gheisari, 2010, 2011; Bakule and Morenzoni, 2004). Naturally, during mean life time of muon, all of these processes compete with each other. On this base, the following rate of equations is written for the populations of the particles:

dN 0 (t ) dt

 0 N 0 (t )  0a N 0 (t )

(9)

dN 0 p (t )  0 N 0 p (t )  c p 00 a N 0 (t )  ct 0  pt N 0 p (t )  c p 00 pp N 0 p (t ) dt dN 0 t (t )  0 N 0 t (t )  ct 0 a N 0 (t )  c p 00  pt N 0 t (t )  ct 0  pt N 0 p (t ) dt  ct 0  tt N 0 t (t ) dN 0 tt (t )  0 N 0 tt (t )  f tt N 0 tt (t )  ct 0 tt N 0 t (t ) (12) dt

dt

(11)

dN 0 pt (t )  0 N 0 pt (t )  f pt N 0 pt (t )  c p0 pt N 0 t (t ) (13) dt

dN dt (t ) tr f  0 N dt (t )  cd res dt N t (t )  dt N dt (t ) dt dN p (t )

(10)

(14)

 0 N p (t )  c p a N  (t )  c p pp N p (t )  cd pd N p (t )

(15)

dN d1 / 2 (t )

1 1  cd a N  (t )  cd pd N p (t )  cd1/dd2 N d1 / 2 (t )  cd d N d 3 / 2 (t ) dt 3 3  c p pd N d1 / 2 (t )  0 N d1 / 2 (t )

(16)

dN d 3 / 2 (t )

2 2 /2  c d  a N  (t )  c d  pd N p (t )  c d  3dd N d 3/ 2 (t )  c p pd N d 3 / 2 (t ) dt 3 3  c d  d N d 3/ 2 (t )  0 N d 3/ 2 (t ) dN  (t )  0 N  (t )  f dt (1  seff ) N dt (t )  f pt (1   pt ) N pt (t )  aN  (t ) dt  f dd (1  0.58dd ) N dd (t )

(17)

(18)

Avazpour et al.

Table 1. The different values of the muon cycling coefficient at a temperature of about 3K.

DISCUSSION In this work, we have tried to study muon catalyzed fusion in the heterogeneous target H  T / D2 / H  D

c

cp

using incident muons of momentum 26.25 MeV / c . The target has been assumed to be at low temperature. After injection of such fast muons into the H-T layer, a large fraction of t1s atoms leave the layer by help of RT

0.001 87.5 0.01 87 0.1 80 0.2 72 0.25 68 0.3 64 0.4 55 0.5 45 0.6 36 0.7 27 0.75 22 0.8 17.5 0.85 12.6 0.9 8.8 0.98 1.8 0.99 0.88 0.995 0.46 The notation  c represents the muon cycling

effect. The RT effect has significantly been observed in solid hydrogen targets adopted with a small amount of tritium. We have taken ct  0.12% for the composition of the first layer.

d  (t ) dt

t1s

atoms

D2 layer, dt molecules are formed resonantly in the H  D layer. The released muons can form other muonic molecules, leading to fusion. The leakage of muons from the condensed H/D layer is low. On this base, we used only three hydrogen layers. The rate of Equations 9 to 21 has been written for description of such processes in muon catalyzed fusion. Our equations have been solved by using Lsode computer code for different deuterium concentrations in the last layer. Our results show that the muon cycling coefficient (  c ) in the fusion layer

cp dt

After slowing down of

through

coefficient.

dN pt (t )

2285

 0 N pt (t )  pt N pt (t )  c p pt N (t ) f

(19)

reaction (Equation 7), the fusion yield of

pd has been

neglected for estimation of  c .

tr t

f  f dt N dt (t )  f pt N pt (t )  dd  N dd (t )

equals 87.5 under condition that we take  0.001 . Due to high muon sticking probability in the

(20)

The maximum value of the muon cycling coefficient in the fusion layer equals 88. The differences between the muon cycling coefficients in a low range of deuterium ( c d  2% ) reduces with increasing proton concentration. This may be by the reason of slow variation of the initial

dN dd (t ) dt

 0 N dd (t )   N dd (t )  cd N d 3 / 2 (t ) (21) f dd

3/ 2 dd

population of the d in the spin-flip interaction. The deceleration of dμ muonic atom below the Debye energy 3/ 2

is very slow, and that mean energy of d atom is always significantly greater than 1 meV in pure deuterium (Adamczak and Faifman, 2005). By this reason, we have used the dd formation rate calculated by Adamczak et al. (1991, 2003). As an error estimation, the validity of the present results is more for low proton concentrations. 3/ 2

tr

where N t (t ) denotes

t1s atoms

a

time

population

of

fast

which enter the third layer (Gheisari, 2011).

The notations

N (0) (t ) and N ( ) (t ) represent the time

populations of muon and muonic atoms/molecules in the first and third layers, respectively. The value of   t  at

t  2.2s denotes the muon cycling coefficient (  c ).

ACKNOWLEDGEMENTS

The results of this coefficient are presented in Table 1. It is necessary to mention that the tritium impurity ct  0.12% , the values of the relative densities (φ0 =

The first author gratefully acknowledges the Yasouj University Research Council. The second author would like to thank the Persian Gulf University Research Council.

1.2, φ = 1.43), and the rates have been extracted from the literature data (Markushin, 1995; Filipowicz et al., 2008; Gheisari, 2011; Adamczak and Faifman, 2005; Mulhauser et al., 2006) and muon sticking probabilities (Mulhauser et al., 2001).

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